Linear-Time Algorithm for Sliding Tokens on Trees
Author(s)Demaine, Erik D.; Demaine, Martin L.; Fox-Epstein, Eli; Hoang, Duc A.; Ito, Takehiro; Ono, Hirotaka; Otachi, Yota; Uehara, Ryuhei; Yamada, Takeshi; ... Show more Show less
Polynomial-Time Algorithm for Sliding Tokens on Trees
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Suppose that we are given two independent sets I [subscript b] and I [subscript r] of a graph such that ∣ I [subscript b] ∣ = ∣ I [subscript r] ∣, and imagine that a token is placed on each vertex in I [subscript b]. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I [subscript b] and I [subscript r] so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we show that the problem is solvable for trees in quadratic time. Our proof is constructive: for a yes-instance, we can find an actual sequence of independent sets between I [subscript b] and I [subscript r] whose length (i.e., the number of token-slides) is quadratic. We note that there exists an infinite family of instances on paths for which any sequence requires quadratic length.
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Algorithms and Computation
Demaine, Erik D., Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada. “Polynomial-Time Algorithm for Sliding Tokens on Trees.” Lecture Notes in Computer Science (2014): 389–400.
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